Help with chi square distribution

[sneaky666]

How do i show that the a [X1 has a chi square distribution with n degrees of freedom] + [X2 has a chi square distribution with m degrees of freedom] is a [X1+X2 has a chi square distribution with n+m degrees of freedom]?

How can i use moment generating functions to do this?

 

[statdad]

Yes.

 

[ych22]

How do i show that the a [X1 has a chi square distribution with n degrees of freedom] + [X2 has a chi square distribution with m degrees of freedom] is a [X1+X2 has a chi square distribution with n+m degrees of freedom]?

How can i use moment generating functions to do this?

M_X1=(1-2t)^-n/2
M_X2=(1-2t)^-m/2

M_X1+X2=M_X1*M_X1= (1-2t)^-(n+m)/2

which is the MGF of a chi-square distribution with (n+m) degrees of freedom.

That's all I can come up with...not terribly good with this...

 

[SW VandeCarr]

How do i show that the a [X1 has a chi square distribution with n degrees of freedom] + [X2 has a chi square distribution with m degrees of freedom] is a [X1+X2 has a chi square distribution with n+m degrees of freedom]?

How can i use moment generating functions to do this?

If you want to show that a given set of random variables has a chi square distribution and that these distributions are additive you need to start with the Gaussian and then derive the characteristic function from the MGF.

http://www.planetmathematics.com/CentralChiDistr.pdf

 

[blue_raver22]

The chi square distribution is a commonly used statistical distribution in many scientific fields, including biology, psychology, and finance. It is often used to analyze categorical data and to test for the independence of two categorical variables.

To show that the sum of two chi square distributions with n and m degrees of freedom respectively, is a chi square distribution with n+m degrees of freedom, we can use the properties of moment generating functions. The moment generating function is a mathematical tool that allows us to calculate the moments of a distribution, including the mean, variance, and higher moments.

We know that the moment generating function of a chi square distribution with n degrees of freedom is given by M(t) = (1-2t)^(-n/2). Similarly, the moment generating function of a chi square distribution with m degrees of freedom is M(t) = (1-2t)^(-m/2).

Now, to show that the sum of two chi square distributions is also a chi square distribution, we can simply multiply their moment generating functions together. This results in M(t) = (1-2t)^(-(n+m)/2), which is the moment generating function of a chi square distribution with n+m degrees of freedom.

Therefore, we can conclude that the sum of two chi square distributions with n and m degrees of freedom respectively, is a chi square distribution with n+m degrees of freedom. This property can be very useful in statistical analysis, as it allows us to easily calculate the moments and other properties of the sum of multiple chi square distributions.

In summary, by using the properties of moment generating functions, we can show that the sum of two chi square distributions is also a chi square distribution, and that the degrees of freedom of the resulting distribution is the sum of the degrees of freedom of the individual distributions. I hope this explanation helps you better understand the chi square distribution and its properties.